How do I include drag-and-drop in a plugin?

WordPress has a few of these libraries that come with core, you can view the list here: Default Scripts Included and Registered by WordPress. What you’re looking for is Draggable, Sortable, and in your case Droppable.

You would use wp_enqueue_script() to add it like so:

wp_enqueue_script( 'jquery-ui-droppable' );

WordPress registers these scripts using a Handle which you can use to add them, the list of Script Handles can be found in the first link above. You would still need to hook into the admin panel and add the script:

function load_custom_wp_admin_style() {
        wp_enqueue_script( 'jquery-ui-droppable' );
}
add_action( 'admin_enqueue_scripts', 'load_custom_wp_admin_style' );